The two materials used in this chapter are the box of sticks and the classified nomenclature of geometry. The box of sticks is the most important instrument used by the teacher for the presentations, and in the succeeding work of the child.
The classified nomenclature codes the concepts given in the presentations. It is a bank of information to which the child will refer constantly.
Etymology continues to play a crucial role, as we take into consideration the different psychological realities of the child from 3 – 6 years old, and the child from 6 – 12 years old. From 3 – 6, the child has a drive to know “things”, while from 6 – 12, the child has a drive to know “the reasons of things”. Therefore at the level of language, the “thing” is given by its name, and the “reason of the thing” is given with the etymology of the name.Materials:
Box of sticks
Accompanying box of supplies
Wall chart of pictures and labels
Description of materials: Box of sticks
Eleven series of sticks, the first ten of ten different colors and ten different lengths; the last is a series of varied natural sticks.
dark brown – 2cm light brown – 12cm
violet – 4cm green – 14cm
orange – 6cm pink – 16cm (lengths are hole to hole)
red – 8cm blue – 18cm
black – 10cm yellow – 20cm
Eight of each of these ten series have holes at the extremities; two of each have holes all along the length. The natural sticks are of ten different lengths. These serve to construct right-angled isosceles triangles whose hypotenuse are equal to s 2. Therefore these sticks have the lengths of 2 2, 4 2, 6 2, 8 2 and so on.
Three groups of semi-circumferences and three circles.
red circle – diameter of 10; corresponding green semi-circumference
silver circle – diameter of 7; corresponding orange semi-circumference
white circle – diameter of 5; corresponding blue semi-circumference
Box of supplies:
four different colors of thumbtacks
red upholstery nails
three crayons – red, blue, black
Classified Nomenclature for Geometry:
Series A is an exception since it also includes an envelope containing two white pieces of paper picturing a point and a line in red; a red square of paper (surface) and a cube constructed of red paper and dismantled to be stored in the envelope. Note: The line goes off the edges of paper to imply infinity.
While most secondary schools present these concepts starting with the point, the most abstract, and progressing to the solid, reality, we will in the elementary school begin with reality; the concept of the body and go on to the surface, line and point. In the second presentation after the child has worked with these concepts we will present them in reverse order: point, line, surface, solid.
A box – parallelepiped, a can-cylinder, a ball – sphere
The Geometric solids and the stands
Decimal system materials: cube, squares, bars and beads
Pencil sharpener, pencil, paper
A. The teacher asks the child to bring one of the three objects to place in the center of the table. Now put another object in the place of that one. It can’t be bone unless the first object is moved. Try with the third object.
We see that we cannot place an object in a place occupied by another object. Everything that occupies a space is called a solid. This box has flat surfaces. The ball has curved surfaces, so it is the opposite. Since the can has flat surfaces and curved surfaces, it can be placed between them. Examine the geometric solids, naming them as they are divided into three groups: 1) cube, square-based parallelepiped, regular triangular prism, square-based pyramid, regular triangular pyramid; 2) cylinder, cone; 3) sphere, ovoid (or ovaloid), ellipsoid.
Touching the objects lightly, the teacher compares the surface to a very thin veil of paint, or to a piece of paper. With an inset or an object, the teacher lightly runs her fingers over the surface: this is the surface. We can touch anything in the environment and label it a surface; table, wall, face, the globe. The concept of surface is infinite. It goes on in all directions to infinity.
The teacher runs a finger along the edge of the box. This is a line, this is another line. continue identifying lines on other objects in the room, on the geometric solids. In reality the line as a concept has only one dimension, and it is infinite in both directions. Let’s try to draw a line. Sharpen the pencil well. Invite the child to draw a line. That’s too thick. Sharpen the pencil more and try again. This is okay, but it’s still too thick. It should have no thickness at all. And it should be much longer. It should to out the door, across the fields, into the woods; and in the other direction as well, past the deck, through the field, down the street.
On the corners of the box, identify a point, another point. What is a point? It is nothing, but it is similar to a grain of sand, a particle of dust, a grain of pollen.
Let’s try to draw one. Be sure the pencil has a sharp point. Invite the child to make a dot. That’s too big, try again. It is still too big. The concept of a point is that element that has no dimension at all. Note: Only the solid has a definition, the others can only be compared to things.
Bring out the special envelope for this series and construct the cardboard cube. The teacher identifies the solid, and the surface. Turn and bend the surface – it’s still a surface. Identify the line and change its position – it’s still a line. Identify the point. The child matches the labels and definitions using the wall chart and control booklet for control.
B. Note: The way in which the quantities of the decimal system were presented is very important now. The identification has already been made between the unit and the point, the bar and line, the square and surface and the cube and solid. The bead had to be held carefully for it is so small it might roll away. the bar gave an idea of length like a cane. The square covered the palm of your hand. The cube filled up the hand so there was room for nothing else.
The teacher places a unit bead on the table. This is a bead, a unit; it is a point. Add another point, still another point… I’ve made a line. The point is the constructor of the line.
Replacing the beads with a bar of ten, the teacher identifies the line. Other lines are added, making a column of bars. I’ve made a surface. The line is the constructor of the surface.
Replacing the bars with a square, the teacher identifies the surface. The squares are stacked; this is another surface, still another surface… I’ve made a solid. The solid is made up of many surfaces. The surface is the constructor of the solid. In reality, the point, therefore, is the constructor of the solid.
Imagine that this point (the bead) is on fire, like the end of an incense stick in a dark room or the phosphorescent plankton at night. Roll the bead. When it moves I see a line. This moving point has made a line before my eyes.
Imagine that this line (bead bar) is on fire. Hold it a one end and quickly move it back and forth. This moving line determines a surface.
Imagine that this surface is on fire. In moving the surfaces I create a solid.
The point is the constructor of reality. Take an object from the environment, a box or can. All things in reality are limited by surfaces. The surfaces, in turn, contain lines. the lines are made up of points, and the point is made up of nothing. Technically the point has no dimensions.
The same identification can be made of unit beads, when placed far away the cube itself appears to be no more than a unit bead.
The work with the classified nomenclature involves the picture cards in the folder, with reading labels and the same definitions.
Direct Aim: To furnish the fundamental concepts: point, line, surface, solid.
Indirect Aim: To furnish the concepts of all plane figures (which are simply surfaces) and all geometric solids (which are simply solids) in preparation for the study of area and volume.
Age: After 6 years